User blog:Boboris02/Boris´s sezration notation
BORIS´S SEZRATION NOTATION Hello,my name is Boris and I´ve allways been facinated by large numbers. So I decided to write this article to show you my own invention,that I like to call "Boris´s sezration notation" because I think the name is cool and I like having my name in something. Let´s start with the simplest form of it:A two-chain Sopose you have two whole possitive integers (a and b). The simplest sezration you can do with it is this:\(a,b\) and that is equal to (in chained arrow notation) a→a→a→....→a and the number of a´s is b Simple,but effective! Now,let´s spice things up a bit with a three-chain! a,b,c=c→c→c→....→c And there´s a,b many c´s. In other words a,b,c=[c,a,b] Okay,but what about a four-chain? a,b,c,d=[d,a,b,c]=d→d→d→d→...→d once again there´s a,b,c many d´s!!! EXAMPLES: 3,2=3→3=27 3,3=3→3→3=3↑↑↑3=tritri 3,4=3→3→3→3 ~ G(G(27)) (using Graham`s number) 3,7=3→3→3→3→3→3→3 3,7,5=5→5→5→5→...→5→5(3,7 many fives) 2,9,3,6=6→6→6...→6→6 (2,9,3 many sixes) THE EXTENDED SEZRATION! If what we did previously wasn`t crazy enuff,I have something new for you! b|a=a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a,a...a,a(where there are b many a`s) EXAMPLES: 4|3=3,3,3,3=[3,[3,[3,3]]=[3,[3,3↑↑↑3]]=[3,3→3→3→...(3↑↑↑3)....→3→3] WOW!!! We can also extend the extention! [5|10|5|4]=[5,5,5,5,5,5,5,5,5,5|4]=4,4,4,4,4,4,.....,4,4 and there are 5,5,5,5,5,5,5,5,5,5 many 4`s! SECOND LEVEL SEZRATION! This one was sugested by AarexWikia04 Sopose you have this |b||a,that equals [a|[a|[a|[a|....a|[b|a]|a]|a]|a....]|a] , with b many pairs of a`s on the sides! EXAMPLE: |4||3=[3|[3|[3|[4|3]|3]|3]|3]=[3|[3|[3,3,3,3|3]|3]|3]=[3|[3,3,3,3,....([3,3,3,3 three`s)....,3,3,3]|3]|3]=[REALY BIG|3]=BIG 3`s!)...3,3,3 THIRD LEVEL SEZRATION! ||b|||a=|[a||[....a||[b||a....]||a]||a] , with b many pairs of a.(No examples for this one) FOURTH LEVEL SEZRATION! |||b||||a=||[a|||[...a|||[b|||a...]|||a]|||a] , with b pairs of a`s! HIGHER DIMENTIONAL SEZRATION! §=dimention of (§a)=dimention of a a(§c)b(§c)a=|||||||||||||....||b|||...|||||||||||||||||||a ,with c pairs of lines!!! If you thought you know,what big means...think again! TETRATIONAL SEZRATION! a(§(§b))c(§(§b))a=[a(§a(§b)c(§b)a)c(§a(§b)c(§b)a)a] In other words: If a(§b)c(§b)a=x ,then a(§(§b))c(§(§b))a=a(§x)c(§x)a PENTATIONAL,HEXATIONAL,HEPTATIONAL... SEZRATION! Pentational-a(§(§(§b)))c(§(§(§b)))a Hexational-a(§(§(§(§b))))c(§(§(§(§b))))a Heptational-a(§(§(§(§(§b)))))c(§(§(§(§(§b)))))a You can probably guess how powerfull these can be and how they work by yourself! UPPER SEZRATION! a/b=a(§(§(...(§a)))...)b(§(§(...(§a)))...)a where there are b many §`s a/b/2=[a/b/a/b] a/b/3=[a/b/2/a/b/2] a/b/4=[a/b/3/a/b/3] a/b/c=[a/b/c-1/a/b/c-1] If a/b/c=x,then a/b/c+1=x/x=z and a/b/c+2=[x/x/x/x]=z/z a/b/c/2=[a/b/c/a/b/c] a/b/c/3=[a/b/c/2/a/b/c/2] a/b/c/d=[a/b/c/d-1/a/b/c/d-1] a//b=a/a/a/a/a/a/..../a,whith b a`s a///b=a//a//a//a//...//a,whith b a`s And so on! ULTRA-SEZRATION! Things like n///n can be expressed like this n/n_{3}(where 3 is a subscript). Now,you can write things like n/n_{100},without having to write them down,but that's not ultra-sezration(yet)! Ultra-sezration occurs,when things like a(@)b appear. a(@)b=a/b_{a/b_{a/b..._{a/b}...} with a/b subscripts of a/b a(@+1)b=[a(@)[a(@)[a(@)[.....a(@)a]]]....]],with b+1 a's a(@+2)b=[a(@+1)[a(@+1)[a(@+1)[...a(@+1)a...] - b+1 a's a(@+@)b=[a(@+[a(@+[a(@+....a(@)a)a])a])a...)a] - b+1 a's a(@+@+@)b or a(3@)b=[a(@+@+[a(@+@+[a(@+@+....a(@+@)a)a])a...])a] And so on! a(@^2)b=[a(@*[a(@*[a(@*[...a(@*b)a)a])a])a.....)a] with a(@*b)a a's a(@^3)b=[a(@^2*[a(@^2*[....a(@^2*b)aa)]a)....a)])a] - a(@^2*b)a a's a(@^@)b=[a(@^[a(@^[a(@^[....a(@^b)a)a])a....])a] - a(@^b)a a's a(@^@^@)b=[a(@^@^[a(@^@^[....a(@^@^b)a)a])a....])a] - a(@^@^b)a a`s a(@↑↑@)b=[a(@↑↑[a(@↑↑[...a(@↑↑b)a)a])a]....)a])a] - a(@↑↑b)a a's a(@↑↑↑@)b=[a(@↑↑↑[a(@↑↑↑[a(@...a(@↑↑↑a(@↑↑↑b)a)a])a])a...])a] - a(@↑↑↑b)a a's a(@↑↑↑↑@)b=[a(@↑↑↑↑[a(@↑↑↑↑[a(@....a(@↑↑↑↑a(@↑↑↑↑b)a)a])a])a....])a] a(@→@→5)b=[a(@→[a(@→[a(@→...a(@→b→5)a→5)a]→5)a]→5)a]...]→5)a] - a(@→b→5)a a's a(@→@→c)b=[a(@→[a(@→[...[a(@→a(@→b→c)a→c)a]→c)a]→c)a]....]→c)a] - a(@→b→c)a a's a(@→@→@→2)b=[a(@→@→[a(@→@→[a(@→@→....[a(@→@→a(@→@→b→2)a→2)a]→2)a]...]→2)a] - a(@→@→b→2)a a's *NEW* (suggested by Chronolegends,Alemagno12,AarexWikia04) a(g_@)b=a(@→@→@→@→@→@→...(@)...→@→2)b~[a(@,@)b] a(g+1_@)b=[a(g_@)[a(g_@)...a(g_@)a]]...]] - a(g_@)b pairs of brackets a(g+2_@)b=[a(g+1_@)[a(g+1_@)...a(g+1_@)a]....] - a(g_@)b pairs of brackets a(g*2_@)b~[a(@,@,@)b] a(g*3_@)b~[a(@,@,@,@)b] a(g*n_@)b~[a(n+1|@)b] a(gg_@)b=[a(g*b_@)[....[a(g*b_@)a(g*b_@)a]...]] - a(g*b_@)a pairs of brackets a(ggg_@)b=[a(gg_@)[a(gg_@)[...a(gg_@)a]...]] - a(gg_@)b pairs of brackets a(№g_@)b=a(ggggg...(b)...ggg_@)a a(g_g_@)b=[a(№g_@)[a(№g_@)[....a(№g_@)a...]] - a(№g_@)b pairs of brackets a(g_g_g_@)b=[a(g_g_@)[a(g_g_@)[....a(g_g_@)a...]] - a(g_g_@)b pairs of brackets a(№(g_g)_@)b=a(g_g_g_g_g_...(b)..._g_g)a a(g__g_@)b=[a(№(g_g)_@)[a(№(g_g)_@)[...[a(№(g_g)_@)a(№(g_g)_@)a)a])a...])a] - a(№(g_g)_@)b pairs of brackets a(g___g_@)b=[a(g__g_@)[a(g__g_@)[...[a(g__g_@)a(g__g_@)a)a])a...])a] - a(g__g_@)b pairs of brackets a(№(_g_)_g_@)b=a(g_________...(b)...___g_@)a a(G_@)b~[a(@(§b)@(§b)@)a] a(G+1_@)b~[a(@(§(§b))@(§(§b))@)a] a(G+2_@)b~[a(@(§(§(§b)))@(§(§(§b)))@)a] n(G+n_@)n~[n(@/n+1n) n(G*2_@)n~[n(@/@/n+1)n] n(G*n_@)n~[n(@/@/@/@...(n).../@/n+1)n]~[n(@//n+1)n] n(G^2_@)n~[n([@//@/n+1])n] n(G^3_@)n~[n([@///@/n+1])n] n(G^c_@)n~[n([@/@/n+1]_{c})n] n(G↑↑2_@)n~[n(@(@)n)n] n(G↑↑3_@)n~[n(@(@^2)n)n] n(G↑↑4_@)n~[n(@(@^@)n)n] n(G↑↑5_@)n~[n(@(@↑³@)n)n] n(G↑↑6_@)n~[n(@(@↑↑↑↑@)n)n] n(G↑↑b_@)n~[n(@(@↑...(b-2)...↑@)n)n] (for b ≥ 5 ) n(G↑³2_@)n~[n(@(@→@→@→2)n)n] n(G↑³3_@)n~[n(@(@→@→@→3)n)n] n(G↑³f_@)n~[n(@(@→@→@→f)n)n] n(G↑↑↑↑f_@)n~[n(@(@→@→@→@→f)n)n] n(G→f→v_@)n~[n(@(@→@→@→...(v)...→@→@→f)n)n]~[n([@(@,v)n])n]~[n(@(g_@)v)n] [n(G,4_@)n] > [n(@(g*n_@)n)n] [n(G,5_@)n] > [n(@(G_@)n)n] [n(G,6_@)n] > [n(@(G+n_@)n)n] [n(G,7_@)n] > [n(@(G*n_@)n)n] [n(G,8_@)n] > [n(@(G^n_@)n)n] [n(G,9_@)n] > [n(@(G↑↑n_@)n)n] [n(G,10_@)n] > [n(@(G↑³n_@)n)n] [n(G,11_@)n] > [n(@(G↑↑↑↑n_@)n)n] [n(G,t_@)n] > [n(@(G↑↑...(t-7)....↑↑n_@)n)n] (for t ≥ 8) [n(G,G_@)n] > [n(@(G→G→G→2_@)n)n] [n(G,2,G_@)n]=[n([G,G,2]_@)n] > [n([@(G,G_@)n])n] [n(G,3,G_@)n]=[n([G,G,3]_@)n] > [n([@(G,2,G_@)n])n] [n(G,r,G_@)n] > [n([@(G,r-1,G_@)n])n] [n(G,G,G_@)n] > [n([@(G,n,G+n_@)n])n] [n(G,G,G,G_@)n] > [n([@(G,G,n,G*n_@)n])n] [n(4|G_@)n] > [n([@(G,G,G,n,G^n_@)n])n] [n(h|G_@)n] > [n([@(G,G,G,..(h-1)...G,n,G↑↑...(h-3)...↑↑h_@)n])n] (for h ≥ 4) [n(|h||G_@)n] > ≈[n([@(h+3|G_@)n])n] [n(||h|||G_@)n] > ≈[n([@(|2h+3||G_@)n])n And so on! COMPERISON: n,n~f_w^2(n) n,n,n~f_w^2(f_w^2(n)) n|n~f_w^2+1(n) |n||n~f_w^2+2(n) ||n|||n~f_w^2+3(n) n(§n)n(§n)n~f_w^2+w(n) n/n~f_w^2+w2(n) n/n/n~f_w^2+w2+1(n) n/n/n/n~f_w^2+w2+2(n) n//n~f_w^2+w3(n) n///n > f_w^2+w4(n) n/n_{n} > f_w^2+w^2(n) n(@)n > f_w^4(n) n(@+1)n > f_w^4+w(n) n(@+2)n > f_w^4+w2(n) n(@+n)n > f_w^4+w^2(n) n(@+@)n > f_w^w+w^4(n) n(@+@+@)n > f_w^{w^w+w^4}(n) n(4@)n > f_w^{w^{w^w+w^4}}}(n) n(@^2)n > f_ε₀(n) n(@^3)n > f_εε₃(n) n(@^@)n > f_φ₂,₀(n) n(@^@^@)n > f_φφ₂,ω(n) n(@↑↑@)n > f_Γφ₂,ω(n) (f_Γ₀(n) is also called f_φ₁,₀,₀(n)) n(@↑↑↑@)n > f_ΓΓΓ...Γφ₂,ω ~f_φ₁,₀,₂,ω(n) n(@↑↑↑↑@)n >> f_φ₁,₁,₀,₂,ω n(@→@→5)n >> f_φ₁,₁,₀,₂,₃,ω(n) n(@→@→@→2)n >> f_ψ(Capital Omega^{Capital Omega^{n-1}})(n) a(g_@)b >> f_ψ(CO^{CO^{CO^{CO^{...(ω)...^{CO}}}....}}})(n) (CO - Capital Omega) And so on! OK,SO WHAT'S NEXT? We went through a lot of new notation and terms here and with the last few parts we pretty much hit the barier to ultra-sezration and probably all sezration.I could extend it furthur,but we got pretty far beyond comprehensive level and it becomes quite tidious at this point.So,what I decided to do is to create a completely new function,that might(or might not) grow faster,than any other computable function. But,I will do it in another blog. Boboris02 (talk) 17:40, October 11, 2016 (UTC)Boboris02Boboris02 (talk) 17:40, October 11, 2016 (UTC) Category:Blog posts